Open Access
2020 On the local pairing behavior of critical points and roots of random polynomials
Sean O’Rourke, Noah Williams
Electron. J. Probab. 25: 1-68 (2020). DOI: 10.1214/20-EJP499


We study the pairing between zeros and critical points of the polynomial $p_{n}(z) = \prod _{j=1}^{n}(z-X_{j})$, whose roots $X_{1}, \ldots , X_{n}$ are complex-valued random variables. Under a regularity assumption, we show that if the roots are independent and identically distributed, the Wasserstein distance between the empirical distributions of roots and critical points of $p_{n}$ is on the order of $1/n$, up to logarithmic corrections. The proof relies on a careful construction of disjoint random Jordan curves in the complex plane, which allow us to naturally pair roots and nearby critical points. In addition, we establish asymptotic expansions to order $1/n^{2}$ for the locations of the nearest critical points to several fixed roots. This allows us to describe the joint limiting fluctuations of the critical points as $n$ tends to infinity, extending a recent result of Kabluchko and Seidel. Finally, we present a local law that describes the behavior of the critical points when the roots are neither independent nor identically distributed.


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Sean O’Rourke. Noah Williams. "On the local pairing behavior of critical points and roots of random polynomials." Electron. J. Probab. 25 1 - 68, 2020.


Received: 21 February 2019; Accepted: 23 July 2020; Published: 2020
First available in Project Euclid: 18 August 2020

zbMATH: 07252694
MathSciNet: MR4136480
Digital Object Identifier: 10.1214/20-EJP499

Primary: 30C15 , 60B10 , 60F05

Keywords: critical points , fluctuations of critical points , i.i.d. zeros , Local law , pairing between roots and critical points , random Jordan curves , random polynomials , Wasserstein distance

Vol.25 • 2020
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